Global Hopf bifurcation for differential equations with state-dependent delay (Q979023)

The paper proves a topological Hopf bifurcation theorem for differential equations with state-dependent delay (SD-DDEs). SD-DDEs are difficult to fit into the general framework of smooth infinite-dimensional dynamical systems because the right-hand side does not satisfy the usual regularity assumptions, for example, local Lipschitz continuity. One approach to tackling this lack of regularity (the one taken in this paper) is the use of degree theory, which requires lower regularity. By the nature of degree theory, the conclusions are weaker than the conclusions in the classical, differentiable setting. The paper applies general degree theory as found in [\textit{W. Krawcewicz} and \textit{J. Wu}, Theory of degrees with applications to bifurcations and differential equations. Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York (1997; Zbl 0882.58001)] to problems of the type \[ \dot x(t)=f(x(t),x(t-\tau(t)),\sigma),\quad \dot \tau(t)=g(x(t),\tau(t),\sigma), \] where \(f\) and \(g\) are twice continuously differentiable in their arguments. First, the paper introduces the general notation necessary to formulate the conditions under which the degree theory can be applied. Then, it proves that, if the formal linearization of the system in an equilibrium \((x_0,\tau_0,\sigma_0)\) has exactly one pair \(\pm i\beta\) of purely imaginary eigenvalues with a non-zero crossing number (the degree-theoretic generalization of the condition that the eigenvalue pair crosses the imaginary axis transversally), then there exists a sequence of non-trivial periodic solutions that converges to \((x_0,\tau_0,\sigma_0)\). Also, the periods converge to \(2\pi/\beta\). The theorem is made global in the sense that (under certain conditions) any sequence of non-trivial periodic solutions that approaches the equilibrium manifold must approach an equilibrium for which the formal linearization has purely imaginary eigenvalues. At the moment the statement of the theorems does not make clear if the sets of non-trivial periodic solutions can be represented as continuous graphs near the Hopf point (and degree theory may not permit one to make this kind of statements). Also, at the moment, several conditions required in the paper are stronger than those required by \textit{M. Eichmann} [PhD dissertation, Justus-Liebig University, Giessen (2006)], which extended the classical Hopf bifurcation theorem to general SD-DDEs.